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EEE 431 Computational Methods in Electrodynamics. Lecture 3 By Dr. Rasime Uyguroglu. Energy and Power. We would like to derive equations governing EM energy and power. Starting with Maxwell’s equation’s:. Energy and Power (Cont.). Apply H. to the first equation and E. to the second:.

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EEE 431 Computational Methods in Electrodynamics

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Eee 431 computational methods in electrodynamics

EEE 431Computational Methods in Electrodynamics

Lecture 3


Dr. Rasime Uyguroglu

Energy and power

Energy and Power

  • We would like to derive equations governing EM energy and power.

  • Starting with Maxwell’s equation’s:

Energy and power cont

Energy and Power (Cont.)

  • Apply H. to the first equation and E. to the second:

Energy and power cont1

Energy and Power (Cont.)

  • Subtracting:

  • Since,

Energy and power cont2

Energy and Power (Cont.)

  • Integration over the volume of interest:

Energy and power cont3

Energy and Power (Cont.)

  • Applying the divergence theorem:

Energy and power cont4

Energy and Power (Cont.)

  • Explanation of different terms:

  • Poynting Vector in

  • The power flowing out of the surface S (W).

Energy and power cont5

Energy and Power (Cont.)

  • Dissipated Power (W)

  • Supplied Power (W)

Energy and power1

Energy and Power

  • Magnetic power (W)

  • Magnetic Energy.

Energy and power cont6

Energy and Power (Cont.)

  • Electric power (W)

  • electric energy.

Energy and power cont7

Energy and Power (Cont.)

  • Conservation of EM Energy

Classification of em problems

Classification of EM Problems

  • 1) The solution region of the problem,

  • 2) The nature of the equation describing the problem,

  • 3) The associated boundary conditions.

1 classification of solution regions

1) Classification of Solution Regions:

  • Closed region, bounded, or open region, unbounded. i.e Wave propagation in a waveguide is a closed region problem where radiation from a dipole antenna is an open region problem.

  • A problem also is classified in terms of the electrical, constitutive properties. We shall be concerned with simple materials here.

2 classification of differential equations

2)Classification of differential Equations

  • Most EM problems can be written as:

  • L: Operator (integral, differential, integrodifferential)

  • : Excitation or source

  • : Unknown function.

Classification of differential equations cont

Classification of Differential Equations (Cont.)

  • Example: Poisson’s Equation in differential form .

Classification of differential equations cont1

Classification of Differential Equations (Cont.):

  • In integral form, the Poisson’s equation is of the form:

Classification of differential equations cont2

Classification of Differential Equations (Cont.):

  • EM problems satisfy second order partial differential equations (PDE).

  • i.e. Wave equation, Laplace’s equation.

Classification of differential equations cont3

Classification of Differential Equations (Cont.):

  • In general, a two dimensional second order PDE:

  • If PDE is homogeneous.

  • If PDE is inhomogeneous.

Classification of differential equations cont4

Classification of Differential Equations (Cont.):

  • A PDE in general can have both:

  • 1) Initial values (Transient Equations)

  • 2) Boundary Values (Steady state equations)

Classification of differential equations cont5

Classification of Differential Equations (Cont.):

  • The L operator is now:

Classification of differential equations cont6

Classification of Differential Equations (Cont.):

  • Examples:

  • Elliptic PDE, Poisson’s and Laplace’s Equations:

Classification of differential equations cont7

Classification of Differential Equations (Cont.):

  • For both cases a=c=1,b=0.

  • An elliptic PDE usually models the closed region problems.

Classification of differential equations cont8

Classification of Differential Equations (Cont.):

  • Hyperbolic PDE’s, the Wave Equation in one dimension:

  • Propagation Problems (Open region problems)

Classification of differential equations cont9

Classification of Differential Equations (Cont.):

  • Parabolic PDE, Heat Equation in one dimension.

  • Open region problem.

Classification of differential equations cont10

Classification of Differential Equations (Cont.):

  • The type of problem represented by:

  • Such problems are called deterministic.

  • Nondeterministic (eigenvalue) problem is represented by:

  • Eigenproblems: Waveguide problems, where eigenvalues corresponds to cutoff frequencies.

3 classification of boundary conditions

3) Classification of Boundary Conditions:

  • What is the problem?

  • Find which satisfies within a solution region R.

  • must satisfy certain conditions on Surface S, the boundary of R.

  • These boundary conditions are Dirichlet and Neumann types.

Classification of boundary conditions cont

Classification of Boundary Conditions (Cont.):

  • 1) Dirichlet B.C.:

  • vanishes on S.

  • 2) Neumann B.C.:

  • i.e. the normal derivative of vanishes on S.

  • Mixed B.C. exits.

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